Getting Started 
Misconception/Error The student is unable to draw a diagram that reflects the parameters of the modeling situation. 
Examples of Student Work at this Level The student:
 Does not recognize the given lengths as a Pythagorean triple and draws a triangle other than a right triangle.
 Does not correctly locate the segment representing the fence.

Questions Eliciting Thinking What is the greatest common factor of 600, 800, and 1000? What kind of triangle is a 345 triangle?
What does the problem say about the location of the fence? 
Instructional Implications Remind the student that 6810 is a Pythagorean triple so 6008001000 is as well, since each of these values is a perfect square multiple of 6, 8, and 10. Be sure the student understands that this means the model of the tract of land should be a right triangle. Ask the student to draw a right triangle that includes the lengths given in the diagram. Review what is said about the location of the fence and guide the student to add a segment to the diagram that represents the fence.
Next, guide the student to represent the length of the fence with a variable (e.g., l). Ask the student to calculate the area of the original triangular region. Then explain that each smaller region must have an area that is half the area of the original region. Ask the student to determine how the area of the triangular halfregion can be found and to define any additional variables needed (e.g., the height of the triangular halfregion, h). Guide the student to write an equation that models the area (e.g., ).
Review similar triangle relationships and guide the student to write a proportion that relates the heights and bases of the original triangular region and the triangular halfregion (e.g., ). Explain that the two equations written can be used to find the length of fence needed. Ensure that the student understands that providing these two equations answers the second question.
Provide additional opportunities to use geometric methods to model and solve design problems. 
Moving Forward 
Misconception/Error The student is unable to identify variables that are needed to model the problem. 
Examples of Student Work at this Level The student draws an appropriate diagram and may label the length of fence with a variable. But the student is unable to determine that the height of the triangular halfregion is also unknown and needed to model the area of this region. 
Questions Eliciting Thinking How can you find the area of the original triangular tract of land?
What do you need to know to find the area of the triangular halfregion?

Instructional Implications If needed, guide the student to represent the length of the fence with a variable (e.g., l). Ask the student to calculate the area of the original triangular region. Then explain that each smaller region must have an area that is half the area of the original region. Ask the student to determine how the area of the triangular halfregion can be found and to define any additional variables needed (e.g., the height of the triangular halfregion, h). Guide the student to write an equation that models the area (e.g., ).
Review similar triangle relationships and guide the student to write a proportion that relates the heights and bases of the original triangular region and the triangular halfregion (e.g., ). Explain that the two equations written can be used to find the length of fence needed. Ensure that the student understands that providing these two equations answers the second question.
Provide additional opportunities to use geometric methods to model and solve design problems. 
Almost There 
Misconception/Error The student is unable to correctly write equations to model the problem. 
Examples of Student Work at this Level The student draws an appropriate diagram and labels both the length of fence and the height of the triangular halfregion with variables. The student may write one equation that can be used to find the length of the fence but is unable to write a second equation.

Questions Eliciting Thinking What is the relationship between the triangular halfregion and the original triangular tract?
What is the relationship between the sides of these two triangles. 
Instructional Implications If needed, ask the student to determine how the area of the triangular halfregion can be found and to define any additional variables needed (e.g., the height of the triangular halfregion, h). Guide the student to write an equation that models the area (e.g., ).
Review similar triangle relationships and guide the student to write a proportion that relates the heights and bases of the original triangular region and the triangular halfregion (e.g., ). Explain that the two equations written can be used to find the length of fence needed. Ensure that the student understands that providing these two equations answers the second question.
Provide additional opportunities to use geometric methods to model and solve design problems. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student draws a diagram such as the following:
and provides a solution such as:
Let l be the length of the fence and h be the height of the triangular half of the tract. The total area of the region is . We want the area of the triangular half to be 120,000 sq.ft., that is, . Since the triangular halfregion is similar to the original triangular region, . So the equations and can be used to solve for l.

Questions Eliciting Thinking Is it possible to partition the tract into two equal area triangles? 
Instructional Implications Ask the student to use the equations he or she wrote to solve for the length of the fence. 